SATHEE: Properties of Triangles 1 Question 5

Properties of Triangles 1 Question 5

5. If the angles $A, B$ and $C$ of a triangle are in an arithmetic progression and if $a, b$ and $c$ denote the lengths of the sides opposite to $A, B$ and $C$ respectively, then the value of the expression $\frac{a}{c} \sin 2 C + \frac{c}{a} \sin 2 A$ is

(2010)

(a) $\frac{1}{2}$

(b) $\frac{\sqrt{3}}{2}$

(c) 1

(d) $\sqrt{3}$

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Answer:

Correct Answer: 5. (d)

Solution:

Since, $A, B, C$ are in AP.

$\Rightarrow 2 B = A + C$ i.e. $∠ B = 60^{\circ}$

$∴ \frac{a}{c} (2 \sin C \cos C) + \frac{c}{a} (2 \sin A \cos A)$

$= 2 k (a \cos C + c \cos A)$

$using, \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = \frac{1}{k}$

$= 2 k (b)$

$= 2 \sin B$ [using $b = a \cos C + c \cos A$ ] $= \sqrt{3}$

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Properties of Triangles 1 Question 5

5. If the angles A,B and C of a triangle are in an arithmetic progression and if a,b and c denote the lengths of the sides opposite to A,B and C respectively, then the value of the expression acsin⁡2C+casin⁡2A is

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5. If the angles $A, B$ and $C$ of a triangle are in an arithmetic progression and if $a, b$ and $c$ denote the lengths of the sides opposite to $A, B$ and $C$ respectively, then the value of the expression $\frac{a}{c} \sin 2 C + \frac{c}{a} \sin 2 A$ is